Summen av vinkler

Gjenoppfrisking av summering av vinkler.

Formler

$\tan\left(\alpha\right)=\frac{\sin\left(\alpha\right)}{\cos\left(\alpha\right)}$

$\sin\left(\theta+\phi\right)=\sin\left(\theta\right)\cos\left(\phi\right)+\cos\left(\theta\right)\sin\left(\phi\right)$

$\tan\left(\theta-\phi\right)=\frac{\tan\left(\theta\right)-\tan\left(\phi\right)}{1+\tan\left(\theta\right)\tan\left(\phi\right)}$

Problem 1

Bevis følgende $\sin\left(75\right)=\frac{1+\sqrt{3}}{2\sqrt{2}}$

$\sin\left(45+30\right)=\sin\left(45\right)\cos\left(30\right)+\cos\left(45\right)\sin\left(30\right)=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{\sqrt{2}\left(\sqrt{3}+1\right)}{4}$

$\frac{\sqrt{2}\left(\sqrt{3}+1\right)}{4}\ \cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{3}+1}{2\sqrt{2}}$

Problem 2

Bevis følgende $\tan\left(15\right)=\frac{\sqrt{3}-1}{\sqrt{3}+1}$

$\tan\left(15\right)=\tan\left(45-30\right)=\frac{\tan\left(45\right)-\tan\left(30\right)}{1+\tan\left(45\right)\tan\left(30\right)}=\frac{\frac{\sin\left(45\right)}{\cos\left(45\right)}-\frac{\sin\left(30\right)}{\cos\left(30\right)}}{1+\frac{\sin\left(45\right)}{\cos\left(45\right)}\cdot\frac{\sin\left(30\right)}{\cos\left(30\right)}}$

$\frac{\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}-\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}}{1+\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\cdot\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}}=\frac{1-\frac{1}{\sqrt{3}}}{1+1\cdot\frac{1}{\sqrt{3}}}$

Multpliserer med kvadratroten av 3.

$\frac{1-\frac{1}{\sqrt{3}}}{1+1\cdot\frac{1}{\sqrt{3}}}=\frac{1\cdot\frac{\sqrt{3}}{\sqrt{3}}-\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}}{1\cdot\frac{\sqrt{3}}{\sqrt{3}}+\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}}=\frac{\frac{\sqrt{3}}{\sqrt{3}}-\frac{\sqrt{3}}{3}}{\frac{\sqrt{3}}{\sqrt{3}}+\frac{\sqrt{3}}{3}}=\frac{\frac{1}{\sqrt{3}}\left(\sqrt{3}-1\right)}{\frac{1}{\sqrt{3}}\left(\sqrt{3}+1\right)}=\frac{\sqrt{3}-1}{\left(\sqrt{3}+1\right)}$

Formler

Problem 1

Problem 2

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